This link seems ambiguous about the origin, but seems to be popularized by TV show host Jack Parr

: : I KID YOU NOT - Catchphrase used
by Jack Paar. Paar, host of the
Tonight Show from 1957 to 1962,
'invented the talk-show format as we
know it: the ability to sit down and
make small talk big,' said Merv
Griffin. 'Even youngsters sent to bed
before Mr. Paar came on parroted his
jaunty catchphrase, 'I kid you not.'
From "He invented late-night talk,
then walked away," an article in the
Herald-Leader, Lexington, Ky., January
28, 2004.

Grammatically both your versions are correct - "kid you" and "kid with you"

Your wording of the question suggests someone is upset about a joke you've played on them, so it's more common to say

I always thought this phrase originated from young goats. Young goats are called, kids. These creatures are very playful. A joke is playful and fun, as are kids. Hence, "kidding" equates to the act of being playful.

used in the same sense as it is in America today, appears in the British novel A Daughter of the Philistines by Leonard Merrick (1897) and in the British novel The Story of the Amulet by Edith Nesbit (1906). All Jack Paar did was invert the word order to get I kid you not.

Merriam-Webster says that kid (used in this way) is a transitive verb, so the correct American usage is

I'm kidding you.

or

I'm not kidding.

Because the verb is transitive,

Can't I kid with you.

would be incorrect because of the with (and it definitely sounds wrong to me). On the other hand, you can say

Can't I kid around with you.

According to the other answers, the expression seems to have died out in Britain.